A New Formulation of Linear Discriminant Analysis for Robust Dimensionality Reduction

Dimensionality reduction is a critical technology in the domain of pattern recognition, and linear discriminant analysis (LDA) is one of the most popular supervised dimensionality reduction methods. However, whenever its distance criterion of objective function uses <inline-formula><tex-math notation="LaTeX">$L_2$</tex-math><alternatives><inline-graphic xlink:href="nie-ieq1-2842023.gif"/></alternatives></inline-formula>-norm, it is sensitive to outliers. In this paper, we propose a new formulation of linear discriminant analysis via joint <inline-formula><tex-math notation="LaTeX">$L_{2,1}$</tex-math><alternatives><inline-graphic xlink:href="nie-ieq2-2842023.gif"/></alternatives></inline-formula>-norm minimization on objective function to induce robustness, so as to efficiently alleviate the influence of outliers and improve the robustness of proposed method. An efficient iterative algorithm is proposed to solve the optimization problem and proved to be convergent. Extensive experiments are performed on an artificial data set, on UCI data sets, and on four face data sets, which sufficiently demonstrates the efficiency of comparing to other methods and robustness to outliers of our approach.